A numerical semigroup \(\mathcal {S}\)  is said to be symmetric if for every \(x \notin \mathcal {S}\) , \(F(\mathcal {S})-x \in \mathcal {S}\) ; it is well-known that symmetric numerical semigroups are exactly those with type equal to 1. Further, we say that \(\mathcal {S}\)  is almost symmetric if for every \(x \notin \mathcal {S}\)  we have either \(F(\mathcal {S})-x \in \mathcal {S}\)  or \(\lbrace x, F(\mathcal {S})-x\rbrace  \subseteq PF(\mathcal {S})\) . Almost symmetric numerical semigroups can have arbitrarily large type (cf. [1]}); however, for fixed small embedding dimensions (up to 4, cf. [2]}, [3]}) the type is bounded. It is not clear if, once the embedding dimension \(e\)  is fixed, there is some sort of restriction which bounds \(t\)  in function of \(e\)  (see Question REF ). This is interesting since, for numerical semigroups in general, [2]} shows that no upper bound exists for the type \(t\)  even if the embedding dimension \(e\)  is 4.
